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A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1–13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.

Agricultural research in the 21st century has become a collaborative effort. Research on crop pests like Spodoptera frugiperda (J.E. Smith), commonly known as the fall armyworm (FAW), can involve international collaboration because it is a pest not only in the southern United States, but also in Latin and South America. Our interest to study the genetic variation of 24 subpopulations of FAW from the southern United States, Mexico, Puerto Rico, Brazil and Argentina required insect collection procedures that preserve the integrity of DNA for molecular genetic analysis. The samples were collected primarily from maize (Zea mays L.), but also included outliers collected from pigweed (Amaranthus sp.), Royal Paulownia (Paulownia tomentosa (Thunb.) Sieb. and Zucc. ex Steud.), lemon tree (Citrus limon (L.) Burm) and Bermuda grass (Cynodon dactylon (L.) Pers.). A common insect preservation technique is to place individual insects in 95% ethanol (ETOH). However, various regulations for shipping and the size of this insect often prevent large sample sizes stored in ETOH from being imported. Genomic DNA from samples preserved in 95% ETOH, lyophilized and fresh insects was extracted and evaluated using DNA quantification and polymerase chain reaction–amplified fragment length polymorphism (PCR–AFLP). All three treatments yielded high-quality/high molecular weight (c. 70–150 μg) DNA. No differences in quality of genomic DNA for AFLP analysis were observed. Lyophilization is a reliable tool to preserve FAW samples, which yields high-quality DNA for use in AFLP genetic analysis.

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